Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-23T14:10:26.911Z Has data issue: false hasContentIssue false
  • English
  • Français

TWIN SQUAREFUL NUMBERS

Part of: Sequences and sets Diophantine equations

Published online by Cambridge University Press:  19 September 2012

TSZ HO CHAN
TSZ HO CHAN*
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (email: tchan@memphis.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

Keywords

squareful numbersPell equationThue equation

MSC classification

Secondary: 11B05: Density, gaps, topology 11D25: Cubic and quartic equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 93 , Issue 1-2 , October 2012 , pp. 43 - 51
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc.

References

[1]Bennett, M. A., ‘Rational approximation to algebraic numbers of small height: the Diophantine equation $\vert ax^n-by^n\vert =1$’, J. reine angew. Math. 535 (2001), 1–49.CrossRefGoogle Scholar
[2]Bombieri, E. and Schmidt, W. M., ‘On Thue’s equation’, Invent. Math. 88(1) (1987), 69–81.CrossRefGoogle Scholar
[3]Estermann, T., ‘Einige Sätze über quadratfeie Zahlen’, Math. Ann. 105 (1931), 653–662.CrossRefGoogle Scholar
[4]Evertse, J. H., Upper Bounds for the Number of Solutions of Diophantine Equations (Math. Centrum, Amsterdam, 1983).Google Scholar
[5]Huxley, M. N., ‘The rational points close to a curve. III’, Acta Arith. 113(1) (2004), 15–30.CrossRefGoogle Scholar
[6]Ingham, A. E., ‘Some asymptotic formulae in the theory of numbers’, J. Lond. Math. Soc. 2 (1927), 202–208.CrossRefGoogle Scholar
[7]Newman, D. J. and Bateman, P. T., ‘Advanced problems and solutions: solutions: 4459’, Amer. Math. Monthly 61 (1954), 477–479.CrossRefGoogle Scholar